CS276: Information Retrieval and Web Search

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• CS276 Information Retrieval and Web Search
• Pandu Nayak and Prabhakar Raghavan
• Lecture 6 Scoring, Term Weighting and the Vector
  Space Model

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Recap of lecture 5

• Collection and vocabulary statistics Heaps and
  Zipfs laws
• Dictionary compression for Boolean indexes
• Dictionary string, blocks, front coding
• Postings compression Gap encoding, prefix-unique
  codes
• Variable-Byte and Gamma codes

collection (text, xml markup etc) 3,600.0
collection (text) 960.0
Term-doc incidence matrix 40,000.0
postings, uncompressed (32-bit words) 400.0
postings, uncompressed (20 bits) 250.0
postings, variable byte encoded 116.0
postings, g-encoded 101.0
MB
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This lecture IIR Sections 6.2-6.4.3

• Ranked retrieval
• Scoring documents
• Term frequency
• Collection statistics
• Weighting schemes
• Vector space scoring

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Ranked retrieval
Ch. 6

• Thus far, our queries have all been Boolean.
• Documents either match or dont.
• Good for expert users with precise understanding
  of their needs and the collection.
• Also good for applications Applications can
  easily consume 1000s of results.
• Not good for the majority of users.
• Most users incapable of writing Boolean queries
  (or they are, but they think its too much work).
• Most users dont want to wade through 1000s of
  results.
• This is particularly true of web search.

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Problem with Boolean searchfeast or famine
Ch. 6

• Boolean queries often result in either too few
  (0) or too many (1000s) results.
• Query 1 standard user dlink 650 ? 200,000 hits
• Query 2 standard user dlink 650 no card found
  0 hits
• It takes a lot of skill to come up with a query
  that produces a manageable number of hits.
• AND gives too few OR gives too many

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Ranked retrieval models

• Rather than a set of documents satisfying a query
  expression, in ranked retrieval, the system
  returns an ordering over the (top) documents in
  the collection for a query
• Free text queries Rather than a query language
  of operators and expressions, the users query is
  just one or more words in a human language
• In principle, there are two separate choices
  here, but in practice, ranked retrieval has
  normally been associated with free text queries
  and vice versa

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Feast or famine not a problem in ranked retrieval
Ch. 6

• When a system produces a ranked result set, large
  result sets are not an issue
• Indeed, the size of the result set is not an
  issue
• We just show the top k ( 10) results
• We dont overwhelm the user
• Premise the ranking algorithm works

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Scoring as the basis of ranked retrieval
Ch. 6

• We wish to return in order the documents most
  likely to be useful to the searcher
• How can we rank-order the documents in the
  collection with respect to a query?
• Assign a score say in 0, 1 to each document
• This score measures how well document and query
  match.

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Query-document matching scores
Ch. 6

• We need a way of assigning a score to a
  query/document pair
• Lets start with a one-term query
• If the query term does not occur in the document
  score should be 0
• The more frequent the query term in the document,
  the higher the score (should be)
• We will look at a number of alternatives for this.

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Take 1 Jaccard coefficient
Ch. 6

• Recall from Lecture 3 A commonly used measure of
  overlap of two sets A and B
• jaccard(A,B) A n B / A ? B
• jaccard(A,A) 1
• jaccard(A,B) 0 if A n B 0
• A and B dont have to be the same size.
• Always assigns a number between 0 and 1.

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Jaccard coefficient Scoring example
Ch. 6

• What is the query-document match score that the
  Jaccard coefficient computes for each of the two
  documents below?
• Query ides of march
• Document 1 caesar died in march
• Document 2 the long march

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Issues with Jaccard for scoring
Ch. 6

• It doesnt consider term frequency (how many
  times a term occurs in a document)
• Rare terms in a collection are more informative
  than frequent terms. Jaccard doesnt consider
  this information
• We need a more sophisticated way of normalizing
  for length
• Later in this lecture, well use
• . . . instead of A n B/A ? B (Jaccard) for
  length normalization.

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Recall (Lecture 1) Binary term-document
incidence matrix
Sec. 6.2
Each document is represented by a binary vector ?
0,1V
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Term-document count matrices
Sec. 6.2

• Consider the number of occurrences of a term in a
  document
• Each document is a count vector in Nv a column
  below

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Bag of words model

• Vector representation doesnt consider the
  ordering of words in a document
• John is quicker than Mary and Mary is quicker
  than John have the same vectors
• This is called the bag of words model.
• In a sense, this is a step back The positional
  index was able to distinguish these two
  documents.
• We will look at recovering positional
  information later in this course.
• For now bag of words model

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Term frequency tf

• The term frequency tft,d of term t in document d
  is defined as the number of times that t occurs
  in d.
• We want to use tf when computing query-document
  match scores. But how?
• Raw term frequency is not what we want
• A document with 10 occurrences of the term is
  more relevant than a document with 1 occurrence
  of the term.
• But not 10 times more relevant.
• Relevance does not increase proportionally with
  term frequency.

NB frequency count in IR
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Log-frequency weighting
Sec. 6.2

• The log frequency weight of term t in d is
• 0 ? 0, 1 ? 1, 2 ? 1.3, 10 ? 2, 1000 ? 4, etc.
• Score for a document-query pair sum over terms t
  in both q and d
• score
• The score is 0 if none of the query terms is
  present in the document.

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Document frequency
Sec. 6.2.1

• Rare terms are more informative than frequent
  terms
• Recall stop words
• Consider a term in the query that is rare in the
  collection (e.g., arachnocentric)
• A document containing this term is very likely to
  be relevant to the query arachnocentric
• ? We want a high weight for rare terms like
  arachnocentric.

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Document frequency, continued
Sec. 6.2.1

• Frequent terms are less informative than rare
  terms
• Consider a query term that is frequent in the
  collection (e.g., high, increase, line)
• A document containing such a term is more likely
  to be relevant than a document that doesnt
• But its not a sure indicator of relevance.
• ? For frequent terms, we want high positive
  weights for words like high, increase, and line
• But lower weights than for rare terms.
• We will use document frequency (df) to capture
  this.

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idf weight
Sec. 6.2.1

• dft is the document frequency of t the number of
  documents that contain t
• dft is an inverse measure of the informativeness
  of t
• dft ? N
• We define the idf (inverse document frequency) of
  t by
• We use log (N/dft) instead of N/dft to dampen
  the effect of idf.

Will turn out the base of the log is immaterial.
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idf example, suppose N 1 million
Sec. 6.2.1
term dft idft
calpurnia 1
animal 100
sunday 1,000
fly 10,000
under 100,000
the 1,000,000
There is one idf value for each term t in a
collection.
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Effect of idf on ranking

• Does idf have an effect on ranking for one-term
  queries, like
• iPhone
• idf has no effect on ranking one term queries
• idf affects the ranking of documents for queries
  with at least two terms
• For the query capricious person, idf weighting
  makes occurrences of capricious count for much
  more in the final document ranking than
  occurrences of person.

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Collection vs. Document frequency
Sec. 6.2.1

• The collection frequency of t is the number of
  occurrences of t in the collection, counting
  multiple occurrences.
• Example
• Which word is a better search term (and should
  get a higher weight)?

Word Collection frequency Document frequency
insurance 10440 3997
try 10422 8760
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tf-idf weighting
Sec. 6.2.2

• The tf-idf weight of a term is the product of its
  tf weight and its idf weight.
• Best known weighting scheme in information
  retrieval
• Note the - in tf-idf is a hyphen, not a minus
  sign!
• Alternative names tf.idf, tf x idf
• Increases with the number of occurrences within a
  document
• Increases with the rarity of the term in the
  collection

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Score for a document given a query
Sec. 6.2.2

• There are many variants
• How tf is computed (with/without logs)
• Whether the terms in the query are also weighted

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Binary ? count ? weight matrix
Sec. 6.3
Each document is now represented by a real-valued
vector of tf-idf weights ? RV
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Documents as vectors
Sec. 6.3

• So we have a V-dimensional vector space
• Terms are axes of the space
• Documents are points or vectors in this space
• Very high-dimensional tens of millions of
  dimensions when you apply this to a web search
  engine
• These are very sparse vectors - most entries are
  zero.

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Queries as vectors
Sec. 6.3

• Key idea 1 Do the same for queries represent
  them as vectors in the space
• Key idea 2 Rank documents according to their
  proximity to the query in this space
• proximity similarity of vectors
• proximity inverse of distance
• Recall We do this because we want to get away
  from the youre-either-in-or-out Boolean model.
• Instead rank more relevant documents higher than
  less relevant documents

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Formalizing vector space proximity
Sec. 6.3

• First cut distance between two points
• ( distance between the end points of the two
  vectors)
• Euclidean distance?
• Euclidean distance is a bad idea . . .
• . . . because Euclidean distance is large for
  vectors of different lengths.

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Why distance is a bad idea
Sec. 6.3

• The Euclidean distance between q
• and d2 is large even though the
• distribution of terms in the query q and the
  distribution of
• terms in the document d2 are
• very similar.

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Use angle instead of distance
Sec. 6.3

• Thought experiment take a document d and append
  it to itself. Call this document d'.
• Semantically d and d' have the same content
• The Euclidean distance between the two documents
  can be quite large
• The angle between the two documents is 0,
  corresponding to maximal similarity.
• Key idea Rank documents according to angle with
  query.

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From angles to cosines
Sec. 6.3

• The following two notions are equivalent.
• Rank documents in decreasing order of the angle
  between query and document
• Rank documents in increasing order of
  cosine(query,document)
• Cosine is a monotonically decreasing function for
  the interval 0o, 180o

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From angles to cosines
Sec. 6.3

• But how and why should we be computing
  cosines?

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Length normalization
Sec. 6.3

• A vector can be (length-) normalized by dividing
  each of its components by its length for this
  we use the L2 norm
• Dividing a vector by its L2 norm makes it a unit
  (length) vector (on surface of unit hypersphere)
• Effect on the two documents d and d' (d appended
  to itself) from earlier slide they have
  identical vectors after length-normalization.
• Long and short documents now have comparable
  weights

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cosine(query,document)
Sec. 6.3
Dot product
qi is the tf-idf weight of term i in the query di
is the tf-idf weight of term i in the
document cos(q,d) is the cosine similarity of q
and d or, equivalently, the cosine of the angle
between q and d.
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Cosine for length-normalized vectors

• For length-normalized vectors, cosine similarity
  is simply the dot product (or scalar product)
• for q, d
  length-normalized.

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Cosine similarity illustrated
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Cosine similarity amongst 3 documents
Sec. 6.3

• How similar are
• the novels
• SaS Sense and
• Sensibility
• PaP Pride and
• Prejudice, and
• WH Wuthering
• Heights?

term SaS PaP WH
affection 115 58 20
jealous 10 7 11
gossip 2 0 6
wuthering 0 0 38
Term frequencies (counts)
Note To simplify this example, we dont do idf
weighting.
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3 documents example contd.
Sec. 6.3

• Log frequency weighting

• After length normalization

term SaS PaP WH
affection 3.06 2.76 2.30
jealous 2.00 1.85 2.04
gossip 1.30 0 1.78
wuthering 0 0 2.58
term SaS PaP WH
affection 0.789 0.832 0.524
jealous 0.515 0.555 0.465
gossip 0.335 0 0.405
wuthering 0 0 0.588
cos(SaS,PaP) 0.789 0.832 0.515 0.555
0.335 0.0 0.0 0.0 0.94 cos(SaS,WH)
0.79 cos(PaP,WH) 0.69
Why do we have cos(SaS,PaP) gt cos(SaS,WH)?
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Computing cosine scores
Sec. 6.3
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tf-idf weighting has many variants
Sec. 6.4
Columns headed n are acronyms for weight
schemes.
Why is the base of the log in idf immaterial?
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Weighting may differ in queries vs documents
Sec. 6.4

• Many search engines allow for different
  weightings for queries vs. documents
• SMART Notation denotes the combination in use in
  an engine, with the notation ddd.qqq, using the
  acronyms from the previous table
• A very standard weighting scheme is lnc.ltc
• Document logarithmic tf (l as first character),
  no idf and cosine normalization
• Query logarithmic tf (l in leftmost column), idf
  (t in second column), no normalization

A bad idea?
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tf-idf example lnc.ltc
Sec. 6.4
Document car insurance auto insurance Query
best car insurance
Term Query Query Query Query Query Query Document Document Document Document Prod
tf-raw tf-wt df idf wt nlize tf-raw tf-wt wt nlize
auto 0 0 5000 2.3 0 0 1 1 1 0.52 0
best 1 1 50000 1.3 1.3 0.34 0 0 0 0 0
car 1 1 10000 2.0 2.0 0.52 1 1 1 0.52 0.27
insurance 1 1 1000 3.0 3.0 0.78 2 1.3 1.3 0.68 0.53
Exercise what is N, the number of docs?
Score 000.270.53 0.8
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Summary vector space ranking

• Represent the query as a weighted tf-idf vector
• Represent each document as a weighted tf-idf
  vector
• Compute the cosine similarity score for the query
  vector and each document vector
• Rank documents with respect to the query by score
• Return the top K (e.g., K 10) to the user

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Resources for todays lecture
Ch. 6

• IIR 6.2 6.4.3
• http//www.miislita.com/information-retrieval-tuto
  rial/cosine-similarity-tutorial.html
• Term weighting and cosine similarity tutorial for
  SEO folk!